Suppose you are given a quadratic equation with vertex (h, k) and y-intercept, c.
The vertex form of a quadratic equation is given by:
[tex]y=a(x-h)^2+k=ax^2-2ahx+ah^2+k[/tex]
y-intercept is the value of y when x is 0.
Thus, given a y-intercept, c, we have:
[tex]c=ah^2+k \\ \\ \Rightarrow ah^2=c-k \\ \\ \Rightarrow a= \frac{c-k}{h^2} [/tex]
Therefore, given a vertex (h, k) and y-intercept, c, the quadratic equation is given by
[tex]y=\left(\frac{c-k}{h^2}\right)x^2-2\left(\frac{c-k}{h^2}\right)hx+\left(\frac{c-k}{h^2}\right)h^2+k \\ \\ =\left(\frac{c-k}{h^2}\right)x^2-2\left(\frac{c-k}{h}\right)x+c-k+k \\ \\ =\bold{\left(\frac{c-k}{h^2}\right)x^2-2\left(\frac{c-k}{h}\right)x+c}[/tex]
